Bürkner 2021 - Worked Examples Ordinal and Response-Time Models
Summary
Treating VerbAgg responses as three-category ordinal (graded response model) increases precision for high-ability persons; free per-item thresholds are not supported by LOO (, SE 14.1); a partial credit model with category-specific effects shows trait anger matters more for perhaps vs yes than for no vs perhaps. Response-time and drift-diffusion models of a mental-rotation test show rotation lowers drift rate and raises non-decision time but not boundary separation.
Overview
Section 6 compares brms with eRm, ltm, TAM, mirt, sirt, lme4, lavaan, blavaan, MCMCglmm. Section 7 concludes.
Main Content
5.2 Ordinal models (VerbAgg, resp = no/perhaps/yes)
- GRM:
resp ~ 1 + (1 | item) + (1 | id),cumulative+ logit: overall thresholds shared, item-specific shifts. Person estimates agree with the binary 1PL but are more precise for large abilities because yes vs perhaps are kept separate (Fig. 9). - Free thresholds per item:
resp | thres(gr = item) ~ 1 + (1 | id), non-hierarchical priors required (order constraint). (SE 14.1) favours the simpler shifted-threshold model. - Generalised GRM (2PL for ordinal):
disc ~ 1 + (1|i|item)withconstant(1)on the person SD; discrimination does not vary across items (Fig. 10). - Covariates in the GRM: (95% CI ) vs (). Plot category probabilities, not scores (equidistant categories are doubtful).
- Category-specific effects need the PCM (
acat,cs(Anger)), since GRM category-specific effects can imply negative probabilities (Bürkner & Vuorre 2019): () for no vs perhaps; () for perhaps vs yes.
5.3 Response-time models (rotation data; diffIRT; 121 subjects x 10 items)
Items show two 3D objects rotated by 50, 100 or 150 degrees; response same/different (correct/incorrect) and time in seconds recorded.
- Exgaussian:
time ~ rotate + (1|p|person) + (1|i|item)plussigma ~ ...andbeta ~ ...with log links,adapt_delta = 0.99. Mean and spread increase with rotation; skewness stays about constant (consistent with Wagenmakers & Brown 2007). - Three-parameter Wiener drift diffusion (ability test; person and item parameters positive and combined multiplicatively, i.e. additively on the log scale; bias fixed at 0.5):
time | dec(resp) ~ rotate + ...,bs ~ ...,ndt ~ ...,brmsfamily("wiener","log", link_bs="log", link_ndt="log"), with manual inits (ndtintercept ). Drift rate decreases slightly from 50 to 100 degrees and then is flat; non-decision time increases from 50 to 100; boundary separation unaffected, with small item SD (0.05, 95% CI ).
6 Package comparison (Table 3 features)
| Package | Model scope |
|---|---|
| eRm | conditional ML; 1PL and PCM only |
| ltm, TAM, mirt | marginal ML; binary/categorical/ordinal, discrimination, 3PL/4PL; mirt most flexible |
| sirt | collection of special-purpose IRT functions |
| lavaan / blavaan | SEM; multidimensional 1PL/2PL, some ordinal; blavaan Bayesian (MH-Gibbs) |
| lme4 | multilevel; no 2PL, ordinal or categorical |
| MCMCglmm | Bayesian multilevel; no non-linear 2PL |
| brms | Bayesian (adaptive HMC); widest families, non-linear, distributional, multilevel, splines, mixtures, custom families, joint models |
| The IRT packages ship DIF tests; psychotree adds recursive-partitioning DIF. brms needs more setup and has a steeper learning curve. |
7 Conclusion
Beyond flexibility, Bayesian IRT replaces binary decisions based on p-values with probabilities and averaging over uncertainty.
Examples
Reading category-specific effects for marketing surveys
Setup. A 3-point purchase-intent item (no/maybe/yes) and a covariate (ad exposure frequency). Solution. Fit
acatwithcs(exposure). Interpretation. If exposure moves maybe to yes but not no to maybe, a GRM with one coefficient would average over the two transitions and mislead.
Connections
Ordinal-category models relate to ordered logit in Discrete Choice Models; sequential-sampling (Wiener) models are decision models for response-time data.